We study generalized traveling front solutions of reaction-diffusion equations modeling flame propagation in combustible media. Although the case of periodic media has been studied extensively, until very recently little has been known for general disordered media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in this framework.

Thurston asked a bold question of whether finite volume hyperbolic 3-manifolds might always admit a finite-sheeted cover which fibers over the circle. This talk will review some of the progress on this question, and discuss its relation to other questions including subgroup separability, the behavior of Heegaard genus in finite-sheeted covers, CAT(0) cubings, and subgroups of right-angled reflection groups. Some applications of the techniques will also be mentioned.

We study ergodic uniformly discrete point processes with pure point spectrum. A part of the point spectrum is known as Bragg peaks. It can (in principle) be measured in diffraction experiments. We show that such point processes are determinded by finitely many moments whenever the group of eigenvalues is finitely generated over the set of Bragg peaks. In particular, these processes are determined by their first three moments if the set of Bragg peaks agrees with the point spectrum. (The talk is based on joint work with Robert V. Moody)